On inhomogeneous minima of indefinite binary quadratic forms (Q1182368)

The authors extend some asymmetric inequalities due to Blaney and Barnes and Swinnerton-Dyer. They obtain exact values of \(D_ m\) for \(3\leq m\leq 3.9437... \) (Theorem 1) improving those of Blaney, and Dumir and Grover. In Theorem 2 they obtain some lower bounds of \(D_ m\) for \(m\geq 4\) which are better than Blaney's. For \(m\in [3, 22/7]\) they find the first isolation, i.e. the admissible lattice \(L\) is not equivalent to a special lattice \(L_ 0\), then \(D_ m\geq 4(9+7\sqrt 3)m/33\) (Theorem 3) and also observe that the second isolation is not possible for \(3\leq m\leq 22/7\).